1433 Lecture 3 - theadamabrams.com

Math 1433Monday, 18 October

Warm-up: List all the complex numbers that

satisfy .z

z4 = 1

Do this now.

Roots of unityWhat numbers satisfy ?

Answer: 1, -1

What are all the complex numbers that satisfy ?

Note that z2 = 1 or z2 = -1.

Answer: 1, -1, i, -i

z z2 = 1

z z4 = 1

Roots of unityWhat are all the complex numbers that satisfy ?z z3 = 1

z3 = (reØi)3 = r3e(3Ø)i = 1e(0°)i

r3 = 1 and 3Ø = 0° or 360° or 720°

r = 1 Ø = 0° or 120° or 240°same as 0°

or 1080° …

or 360° …

z = 1e(0°)i = 0z = 1e(120°)i = -1/2 + 3/2 iz = 1e(240°)i = -1/2 – 3/2 i

+360°, so same argument

Roots of unityFor any natural number , the solutions to are exactly

.

These are called the nth roots of unity.

n zn = 1z = e(2π/n)i

z = e2⋅(2π/n)i

z = e3⋅(2π/n)i

⋮z = e(n−1)⋅(2π/n)i

z = en⋅(2π/n)i = e2πi = 1

-1.5 -1.0 -0.5 0.5 1.0 1.5

Real solutions to x = 1

-1.5 -1.0 -0.5 0.5 1.0 1.5

Real solutions to x2 = 1

-1.5 -1.0 -0.5 0.5 1.0 1.5


-1.5 -1.0 -0.5 0.5 1.0 1.5


-1.5 -1.0 -0.5 0.5 1.0 1.5


-1.5 -1.0 -0.5 0.5 1.0 1.5


-1.5 -1.0 -0.5 0.5 1.0 1.5


-1.5 -1.0 -0.5 0.5 1.0 1.5


Real numbers are quite boring here.

-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0

1.5Solutions to z5 = 1

-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0


-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0


-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0


-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0

1.5Solutions to z = 1

-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0


-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0


-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.5

1.0


e2( 2π3 )i = −1

2 −3

2 i

e( 2π3 )i = −1

2 +3

2 i

e3( 2π3 )i = 1

Complex numbers are better!

Real vs. complexIn some ways, real numbers are better.

Physical measurementsOrdered: always

In some ways, complex #s are better.

roots – always exactly of themRotation and trig functionsPolynomials — …

x < y or x ≥ y

nth n

The Fundamental Theorem of

Algebra (ver. 1)For any non-constant polyno-mial , there is at least one complex solution to .

f(x)f(x) = 0Not true for real

(example: x2+1 = 0).

PolynomialsA polynomial in the variable is a function of real numbers that can be described by an expression of the form

where is an integer and the emoji are real or complex numbers (called the coefficients).

A real polynomial is one where every coefficient is a real number.

A complex polynomial is one where every coefficient is complex.Real numbers are complex numbers ( ), so every real polynomial is also a complex polynomial.

x

n ≥ 0

a + 0i

😀 + 🧐 + ⋯ + 😂 + 🥺 + 😐,xn xn−1 x2 x

PolynomialsExamples of polynomials:

Examples that are not polynomials:

5x3 − 27x+ 32

82x5 − 9x(x − 1)3

x

x−3

5x2 + 3 + x−1

sin(x)

This can be written as x3+3x2+3x+1, so it is a polynomial.

if the variable is if the variable is

ax + b x7t2 − 8t + 1 t

Roots or zeroesThe zeroes of a polynomial are the values of the inputs for which the output is zero. They are also called roots of the polynomial.

Often, we are interested in particular types of numbers as zeroes.Example: has

Integer root:

Rational roots:

Real roots:

2x6−3x5−21x4+56x3−26x2−245x+525−3

−3 and 52

−3, 52 , 7, and − 7

Complex roots: −3, 52 , 7, − 7, 1+2i, and 1−2i


The Fundamental Theorem of Algebra (ver. 1)

Every non-constant complex polynomial has at least one root.


We often use the variable when we care about complex roots.For example,

“What are the zeroes of ?”Depending who you ask, the answer could be either “ ” or “none” (there are no zeroes).

“What are the zeroes of ?”Answer: .

z

x2 + 1i and −i

z2 + 1i and −i

Finding roots by handExample: Find all roots of .z2 + (1+i)z + i

DegreeThe terms of a polynomial are the expressions that are added or subtracted together.

Example: has four terms:Term one: Term two: Term three: Term four:

The order of the terms doesn’t matter.Some people prefer to write .

f(x) = x5 + 6x3 − 4x + 8x5

6x3

−4x8

8 − 4x + 6x3 + x5

DegreeThe degree of a polynomial is the highest power of the variable that appears in the polynomial. We write for the degree of .

A polynomial is called monic if its highest-degree term has coefficient .

Example:

deg( f ) f(x)

1x3 − 2

5 x + 8

“constant” “linear”*

“quadratic” “cubic”

“quartic”

Degree 0 example: Degree 1 example: Degree 2 example: Degree 3 example: Degree 4 example:

9x + 22x2 − 5x − 12−8x3

x4 − 7x + 1

+ - x ÷We can add two polynomials.

We can subtract two polynomials.

We can multiply two polynomials.

We can try to divide two polynomials, but sometimes the result is not a polynomial (for example, is not a polynomial).1/x

(4x2 − 3x) + (x3 + x2 + 3x + 8) = x3 + 5x2 + 8

(4x2 − 3x) − (x3 + x2 + 3x + 8) = −x3 + 3x2 − 6x − 8

(4x2 − 3x)(x3 + x2 + 3x + 8) = 4x5 + x4 + 9x3 + 23x2 − 24x

+ - x ÷Question: What can we say about and ?deg( f + g) deg( f ⋅ g)

(4x2 − 3x) + (x3 + x2 + 3x + 8) = x3 + 5x2 + 8(4x2 − 3x) + (−4x2 + 7) = −3x + 7

(4x2 − 3x)(x3 + x2 + 3x + 8) = 4x5 + x4 + 9x3 + 23x2 − 24x

is the maximum of .deg( f + g) ≤ deg( f ) and deg(g)

= 4x2(x3+x2+3x+8) + (-3x)(x3+x2+3x+8) = (4x5 + ⋯) + (-3x4 + ⋯)

= exactly.deg( f ⋅ g) deg( f ) + deg(g)xa ⋅ xb = xa+b

FactoringNatural numbers can be “factored” (re-written as a product of smaller numbers).

Example: If , we say that is a factor of .

A natural number other than that cannot be factored is called a prime number. The first several primes are …

We can uniquely factor a natural number as a product of primes.Example:

(If we expand from naturals to integers, we might need to include .)Example:

198 = 6 ⋅ 33a = b ⋅ c b a

12, 3, 5, 7, 11, 13,

198 = 2 ⋅ 32 ⋅ 11−1

−1625 = −1 ⋅ 53 ⋅ 13

Factoring

x2+8x = x(x+8)x2+ 1

2 x = x(x+ 12 )

x3−12x2+41x−42 = (x2−5x+6)(x−7)x3−11x2+34x−42 = (x2−4x+6)(x−7)

If , we say that is a factor of .f(x) = g(x) ⋅ h(x) g(x) f(x)

Polynomials can also be factored. Examples:

Factoring

If , we say that is a factor of .f(x) = g(x) ⋅ h(x) g(x) f(x)Polynomials can also be factored.

The Factor TheoremIf is a factor of the polynomial , then is a zero of .If is a zero of the polynomial , then is a factor of .

(x − r) f(x) r f(x)r f(x) (x − r) f(x)

This means that if we find one zero of —let’s call this number —then the other zeroes of will be zeroes of

. Note that has a lower degree than .

f(x)r f(x)

g(x) = f(x)x − r g f

Finding roots by handExample: Find all roots of , given that is a root.x3 − 13x + 12 3

Slow method: algebra rules (x3 - 13x + 12) = (x-3)(ax2 + bx + c) for some a,b,c = ax3 + bx2 + cx - 3ax2 - 3bx - 3c = ax3 + (b-3a)x2 + (c-3b)x + (-3c) For ax3+⋯ to equal equal 1x3+0x2-⋯, we must have

a = 1, b-3a = 0, c-3b = -13, -3c = 12. This leads to a=1, b=3, c=-4, so

(x3 - 13x + 12) = (x-3)(x2 + 3x - 4). The roots of x2+3x-4 are 1 and -4, so the roots of x3 - 13x + 12 are 1, -4, and 3.

Finding roots by handExample: Find all roots of , given that is a root.x3 − 13x + 12 3

Fast method: “synthetic division”

1 0 -13 12

3

1 0 -13 12

3 3 9 -12

1 3 -4 0

The numbers 1 3 -4 on the bottom row (ignore the 0 for now) tell us that

(x3 - 13x + 12) = (x-3)(1x2 + 3x - 4). As before, the roots of x3-13x+12 are 1, -4, and 3.

Factoring

If , we say that is a factor of .f(x) = g(x) ⋅ h(x) g(x) f(x)Polynomials can also be factored.

Question: How can you tell when you have only irreducible factors?

The answer may depend on whether you allow complex numbers.

= (x−2)(x−3)(x−7)x3−12x2+41x−42 = (x2−5x+6)(x−7)x3−11x2+34x−42 = (x2−4x+6)(x−7)

A polynomial that cannot be factored as a product of non-constant polynomials is called irreducible. Note 2x+10 is irreducible.

2x+10 = 2(x+5) is like -23 = (-1)(23).

1433 Lecture 3 - theadamabrams.com

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